Secondary Isotope Effects on Amplitudes of Molecular Vibrations

Abstract

Secondary isotope effects on amplitudes of bond oscillations in harmonically vibrating polyatomic molecules have been studied using normal coordinate theory. Morino's scheme for circumventing the L matrix by expanding the normal coordinate mean-square amplitude matrix in integral powers of the Λ matrix is extended and found to lead to some simple and useful inequalities. Upper and lower limits of bond amplitudes are established which are rigorously independent of masses of atoms other than the atoms in the bond. It is shown that the upper limit of secondary mass dependency vanishes with increasing temperature much more rapidly than does the primary mass dependency. Even at absolute zero, secondary isotope effects on bond amplitudes are found to be relatively very much smaller than secondary isotope effects on bond frequencies. Numerical results of ``exact'' and various approximate approaches for calculating root-mean-square amplitudes are presented for the C–I bonds in CH3I and CD3I, and for the C=C bonds in C2H4 and C2D4. The smallness of secondary isotope effects on amplitudes indicates that their coupling with Morse bond asymmetry is not a major source of some recently reported secondary isotope effects on bond lengths.